## Spectral elementary-coherence-function representation for partially coherent light pulses

Optics Express, Vol. 15, Issue 8, pp. 5160-5165 (2007)

http://dx.doi.org/10.1364/OE.15.005160

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### Abstract

A broad class of partially coherent non-stationary fields can be expressed in terms of the recently proposed independent-elementary-pulse model. In this work we first introduce a corresponding dual representation in the frequency domain and then extend this concept by considering shifted and weighted elementary spectral coherence functions. We prove that this method, which closely describes practical optical systems, leads to properly defined correlation functions. As an example, we demonstrate that our new model characterizes, in a natural way, trains of ultra-short pulses, affected by noise and timing jitter, emitted by usual modulators employed in telecom applications.

© 2007 Optical Society of America

## 1. Introduction

2. G. P. Agrawal, *Fiber-Optic Communication Systems*, 3rd ed.
(Wiley, New York, NY,
2002). [CrossRef]

4. J. Lancis, V. Torres-Company, E. Silvestre, and P. Andres, “Space-time analogy for partially
coherent plane-wave-type pulses,” Opt.
Lett. **30**, 2973–2975
(2005). [CrossRef] [PubMed]

5. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of
temporally modulated stationary light sources,”
Opt. Express **11**, 1894–1899
(2003). [CrossRef] [PubMed]

2. G. P. Agrawal, *Fiber-Optic Communication Systems*, 3rd ed.
(Wiley, New York, NY,
2002). [CrossRef]

6. M. J. Ablowitz, B. Ilan, and S. T. Cundiff, “Noise-induced linewidth in frequency
combs,” Opt. Lett. **31**, 1875–1877
(2006). [CrossRef] [PubMed]

9. S. T. Cundiff and J. Ye, “Colloquium: Femtosecond optical
frequency combs,” Rev. Mod. Phys. **75**, 325–342
(2003). [CrossRef]

10. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian
pulses,” Opt. Commun. **204**, 53–58
(2002). [CrossRef]

11. P. Vahimaa and J. Turunen, “Independent-elementary-pulse
representation for non-stationary fields,”
Opt. Express **14**, 5007–5012
(2006). [CrossRef] [PubMed]

12. P. Vahimaa and J. Turunen, “Finite-elementary-source model for
partially coherent radiation,” Opt.
Express **14**, 1376–1381
(2006). [CrossRef] [PubMed]

13. J. Capmany “A tutorial on microwave photonic
filters,” J. Lightwave Technol. **24**, 201–229
(2006). [CrossRef]

7. V. Torres-Company, H. Lajunen, and A. T. Friberg, “Effects of partial coherence on
frequency combs,” J. Eur. Opt. Soc. Rapid
Publ. **2**, 07007:1–4
(2007). [CrossRef]

## 2. Correlation functions in terms of elementary coherent pulses

### 2.1. Time-domain approach

_{e}(

*t*

_{1},

*t*

_{2}) =

*a*

^{*}(

*t*

_{1})

*a*(

*t*

_{2})exp[

*i*ω

_{0}(

*t*

_{1}−

*t*

_{2})] is the mutual coherence function of the individual deterministic elementary pulses, and

*g*(

*t*) is a (normalized) real and positive weighting function. The angle brackets in the general definition of the mutual coherence function, shown in Eq. (1), denote an ensemble average over the field realizations. Considering the same pulses in the frequency domain, it can be shown that the corresponding cross-spectral density function takes on the form [11

11. P. Vahimaa and J. Turunen, “Independent-elementary-pulse
representation for non-stationary fields,”
Opt. Express **14**, 5007–5012
(2006). [CrossRef] [PubMed]

*A*(ω) and

*G*(ω) are the Fourier transforms of the functions

*a*(

*t*) and

*g*(

*t*), respectively.

_{2}−ω

_{1}only. As discussed in Ref. [11

11. P. Vahimaa and J. Turunen, “Independent-elementary-pulse
representation for non-stationary fields,”
Opt. Express **14**, 5007–5012
(2006). [CrossRef] [PubMed]

### 2.2. Frequency-domain approach

*W*

_{e}(ω

_{1},ω

_{2}) =

*Ũ*

^{*}

_{e}(ω

_{1})

*Ũ*

_{e}(ω

_{2}), where

*Ũ*

_{e}(ω) =

*A*(ω−ω

_{0}) denotes the Fourier transform of

*U*

_{e}(

*t*). Note that since the complex field is taken coherent, the cross-spectral density is a separable function. Now, a broad class of non-stationary coherence functions can be created by spectrally shifting and weighting the above expression with an arbitrary real and positive function

*W*(ω),

_{N}*Ũ*

_{e}(ω). In this way, the energy spectrum of the non-stationary field [14

14. S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, “Energy spectrum of a nonstationary
ensemble of pulses,” Opt. Lett. **29**, 394–396
(2004). [CrossRef] [PubMed]

*S*(ω), can readily be obtained as

*S*(ω) =

*W*(ω, ω), viz.,

*W*(ω) and the energy spectrum of the coherent pulse. Of course, the spectral degree of coherence can be obtained at once with the definition

_{N}*μ*(ω

_{1},ω

_{2})∣ = 1.

*I*(

*t*) = Γ(

*t*,

*t*) = Γ

*(0)∣*

_{N}*a*(

*t*)∣

^{2}; thus the temporal profile of the waveform is determined solely by the complex envelope of the elementary pulses. We can also directly see that the degree of coherence, defined as the absolute value of

*. This shows that the model introduced in Eq. (3) is valid for all fields that are of the Schell-model form in the time domain, rather than in the frequency domain as in the previous case. We emphasize that these classes of fields are not mutually exclusive and thus in many situations the coherence functions can be represented in either way.*

_{N}*(*

_{N}*t*

_{2}−

*t*

_{1}) that is externally modulated according to a deterministic temporal format

*a*(

*t*). Such a situation appears frequently, for example, in telecom devices [2

2. G. P. Agrawal, *Fiber-Optic Communication Systems*, 3rd ed.
(Wiley, New York, NY,
2002). [CrossRef]

*W*(ω) corresponds to the spectral density (or spectrum) of the light source. In particular, when both the modulation format and the spectral density are Gaussian, Gaussian Schell-model pulses are obtained [5

_{N}5. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of
temporally modulated stationary light sources,”
Opt. Express **11**, 1894–1899
(2003). [CrossRef] [PubMed]

## 3. Spectral elementary-coherence-function representation

*Ũ*

_{pc}(ω) is a field realization of a non-stationary ensemble that is described by the cross-spectral density function

*W*

_{pc}(ω

_{1},ω

_{2}) [15

15. M. Bertolotti, L. Sereda, and A. Ferrari, “Application of the spectral
representation of stochastic processes to the study of nonstationary light
radiation: a tutorial,” JEOS A: Pure
Appl. Opt. **6**, 153 (1997). [CrossRef]

*U*

_{pc}(

*t*) =

*a*

_{pc}(

*t*)exp(−

*i*ω

_{0}

*t*), where

*a*

_{pc}(

*t*) is a random complex field envelope. In the case of pulses, each field realization can be taken to contain a finite energy and thus be square-integrable. Now, using Eq. (10), let us rewrite Eq. (3) as

*S*(ω) =

*W*(ω)⊗

_{N}*S*

_{pc}(ω), where now

*S*

_{pc}(ω) =

*W*

_{pc}(ω,ω). It can be shown that the spectral coherence function obtained by Eq. (11) satisfies the non-negative definiteness condition (see Appendix).

## 4. Practical example

_{0}is the carrier frequency of the laser source and

*N*(

*t*) is a random function that describes its amplitude and phase fluctuations. Further,

*M*(

*t*) = ∑

*ψ(*

_{n}*t*-

*nT*), where ψ(

*t*) denotes the deterministic complex envelope of the pulses and

*T*is the mean period, and

*j*(

*t*) is a real, dimensionless random function of zero mean that describes the timing fluctuations. Now, if the pulse-period fluctuations are small compared with the mean period, we may make a first-order Taylor approximation for the complex field,

*M*[

*t*

_{1}−

*Tj*(

*t*)] ≈

*M*(

*t*) −

*Tj*(

*t*)

*M*(

*t*), where the dot means the time derivative.

*t*

_{1},

*t*

_{2}) = ⟨

*U*

^{*}(

*t*

_{1})

*U*(

*t*

_{2})⟩, by assuming that the noise term

*N*(

*t*) is stationary and taking into account that, since the jitter and the noise originate from different devices, the random processes

*N*(

*t*) and

*j*(

*t*) can be taken as statistically independent. In this way, we find that [8]

*(*

_{N}*t*

_{2}−

*t*

_{1}) = 〈

*N**(

*t*

_{1})

*N*(

*t*

_{2})〉 and Γ

*(*

_{j}*t*

_{1},

*t*

_{2}) = 〈

*j*(

*t*

_{1})

*j*(

*t*

_{2})). The point to emphasize now is that this equation perfectly matches the structure of Eq. (12), by identifying

*= ω*

_{i}*− ω*

_{i}_{0}(

*i*= 1,2), ⊗

_{2}denotes two-dimensional convolution,

*M*͂(ω) is the Fourier transform of

*M*(

*t*), and

*W*is obtained from Γ

_{j}*by the inverse of Eq. (6).*

_{j}*W*(ω) = ∫Γ

_{N}*(τ)exp[−*

_{N}*i*(ω −ω

_{0})τ]

*d*τ. On the other hand, if we consider the case with no jitter, i.e.,

*j*(

*t*) = 0, the elementary coherence functions become fully coherent, corresponding the situations of Section 2.2.

## 5. Summary and conclusions

## Appendix A: Proof of the non-negative definiteness condition

*f*(ω) is an arbitrary, sufficiently well-behaved function [1]. Based on Eqs. (10) and (11) we find that

*W*(ω) is a real and positive function. Thus any function obtained by the representation of Eq. (11) is indeed a proper correlation function. Furthermore, this proof is valid also for the independent-elementary-pulse representation discussed in section 2, since that constitutes a special case (of coherent elementary fields) of the more general approach.

_{N}## Acknowledgments

## References and links

1. | L. Mandel and E. Wolf, |

2. | G. P. Agrawal, |

3. | B. E. A. Saleh and M. I. Irshid, “Collet-Wolf equivalence theorem and
propagation of a pulse in a single-mode optical
fiber,” Opt. Lett. |

4. | J. Lancis, V. Torres-Company, E. Silvestre, and P. Andres, “Space-time analogy for partially
coherent plane-wave-type pulses,” Opt.
Lett. |

5. | H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of
temporally modulated stationary light sources,”
Opt. Express |

6. | M. J. Ablowitz, B. Ilan, and S. T. Cundiff, “Noise-induced linewidth in frequency
combs,” Opt. Lett. |

7. | V. Torres-Company, H. Lajunen, and A. T. Friberg, “Effects of partial coherence on
frequency combs,” J. Eur. Opt. Soc. Rapid
Publ. |

8. | V. Torres-Company, H. Lajunen, and A. T. Friberg, “Coherence theory of noise in ultrashort-pulse trains,” J. Opt. Soc. Am. B (in press). |

9. | S. T. Cundiff and J. Ye, “Colloquium: Femtosecond optical
frequency combs,” Rev. Mod. Phys. |

10. | P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian
pulses,” Opt. Commun. |

11. | P. Vahimaa and J. Turunen, “Independent-elementary-pulse
representation for non-stationary fields,”
Opt. Express |

12. | P. Vahimaa and J. Turunen, “Finite-elementary-source model for
partially coherent radiation,” Opt.
Express |

13. | J. Capmany “A tutorial on microwave photonic
filters,” J. Lightwave Technol. |

14. | S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, “Energy spectrum of a nonstationary
ensemble of pulses,” Opt. Lett. |

15. | M. Bertolotti, L. Sereda, and A. Ferrari, “Application of the spectral
representation of stochastic processes to the study of nonstationary light
radiation: a tutorial,” JEOS A: Pure
Appl. Opt. |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(320.5550) Ultrafast optics : Pulses

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: February 7, 2007

Revised Manuscript: April 12, 2007

Manuscript Accepted: April 12, 2007

Published: April 13, 2007

**Citation**

Ari T. Friberg, Hanna Lajunen, and Victor Torres-Company, "Spectral elementary-coherence-function representation for partially coherent light pulses," Opt. Express **15**, 5160-5165 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-5160

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### References

- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).
- G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, New York, NY, 2002). [CrossRef]
- B. E. A. Saleh and M. I. Irshid, "Collet-Wolf equivalence theorem and propagation of a pulse in a single-mode optical fiber," Opt. Lett. 7, 342-343 (1982). [CrossRef] [PubMed]
- J. Lancis, V. Torres-Company, E. Silvestre, and P. Andres, "Space-time analogy for partially coherent planewave- type pulses," Opt. Lett. 30, 2973-2975 (2005). [CrossRef] [PubMed]
- H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003). [CrossRef] [PubMed]
- M. J. Ablowitz, B. Ilan, and S. T. Cundiff, "Noise-induced linewidth in frequency combs," Opt. Lett. 31, 1875- 1877 (2006). [CrossRef] [PubMed]
- V. Torres-Company, H. Lajunen, and A. T. Friberg, "Effects of partial coherence on frequency combs," J. Eur. Opt. Soc. Rapid Publ. 2, 07007:1-4 (2007). [CrossRef]
- V. Torres-Company, H. Lajunen, and A. T. Friberg, "Coherence theory of noise in ultrashort-pulse trains," J. Opt. Soc. Am. B (in press).
- S. T. Cundiff and J. Ye, "Colloquium: Femtosecond optical frequency combs," Rev. Mod. Phys. 75, 325-342 (2003). [CrossRef]
- P. P¨a¨akk¨onen, J. Turunen, P. Vahimaa, A. T. Friberg, and F.Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002). [CrossRef]
- P. Vahimaa and J. Turunen, "Independent-elementary-pulse representation for non-stationary fields," Opt. Express 14, 5007-5012 (2006). [CrossRef] [PubMed]
- P. Vahimaa and J. Turunen, "Finite-elementary-source model for partially coherent radiation," Opt. Express 14, 1376-1381 (2006). [CrossRef] [PubMed]
- J. Capmany "A tutorial on microwave photonic filters," J. Lightwave Technol. 24, 201-229 (2006). [CrossRef]
- S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, "Energy spectrum of a nonstationary ensemble of pulses," Opt. Lett. 29, 394-396 (2004). [CrossRef] [PubMed]
- M. Bertolotti, L. Sereda, and A. Ferrari, "Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial," JEOS A: Pure Appl. Opt. 6, 153 (1997). [CrossRef]

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